The scaling-law flows: An attempt at scaling-law vector calculus

In this paper, the scaling-law vector calculus, which is connected between the vector calculus and the scaling law in fractal geometry, is addressed based on the Leibniz derivative and Stieltjes integral for the first time. The scaling-law Gauss-Ostrogradsky-like, Stokes-like, and Green-like theorems, and Green-like identities are considered in sense of the scaling-law vector calculus. The scaling-law Navier-Stokes-like equations are obtained in detail. The obtained result is as a potentially mathematical tool proposed to develop an important way of approaching this challenge for the scaling-law flows.